2005/01/14 by M. Mancini, C. Oguey
Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Conformal geometry #Conformal map #Conformal symmetry #Extremal length #Geometry #Invariant (physics) #Laplace transform #Laplace's equation #Logarithm #Materials science #Mathematical analysis #Mathematical physics #Mathematics #Partial differential equation #Power law #Quantum, superfluid, helium dynamics #Surface (topology) #Theoretical and Computational Physics #cond-mat.soft
paper · pdf · doi:10.1140/epje/i2004-10133-x
10 pages, 7 figures
arxiv created 2005/01/14 · openalex publication_date 2005/05/04 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
A liquid foam in contact with a solid surface forms a two-dimensional foam on the surface. We derive the equilibrium equations for this 2D foam when the solid surface is curved and smooth, generalising the standard case of flat Hele Shaw cells. The equilibrium conditions at the vertices in 2D, at the edges in 3D, are invariant by conformal transformations. Regarding the films, conformal invariance only holds with restrictions, which we explicit for 3D and flat 2D foams. Considering foams confined in thin interstices between two non parallel plates, normal incidence and Laplace's law lead to an approximate equation relating the plate profile to the conformal map. Solutions are given for the logarithm and power laws in the case of constant pressure. The paper concludes on a comparison with available experimental data.